Maksimov and Kolovsky, Equation (3)

Percentage Accurate: 73.8% → 87.5%

Time: 7.2s

Alternatives: 14

Speedup: 0.3×

Specification

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = cos((k / 2.0d0))
    code = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Initial Program: 73.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(\frac{K}{2}\right)\\ \left(\left(-2 \cdot J\right) \cdot t\_0\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot t\_0}\right)}^{2}} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (/ K 2.0))))
   (* (* (* -2.0 J) t_0) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) t_0)) 2.0))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((K / 2.0));
	return ((-2.0 * J) * t_0) * sqrt((1.0 + pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    real(8) :: t_0
    t_0 = cos((k / 2.0d0))
    code = (((-2.0d0) * j) * t_0) * sqrt((1.0d0 + ((u / ((2.0d0 * j) * t_0)) ** 2.0d0)))
end function

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((K / 2.0));
	return ((-2.0 * J) * t_0) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * J) * t_0)), 2.0)));
}

Python

def code(J, K, U):
	t_0 = math.cos((K / 2.0))
	return ((-2.0 * J) * t_0) * math.sqrt((1.0 + math.pow((U / ((2.0 * J) * t_0)), 2.0)))

Julia

function code(J, K, U)
	t_0 = cos(Float64(K / 2.0))
	return Float64(Float64(Float64(-2.0 * J) * t_0) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * J) * t_0)) ^ 2.0))))
end

MATLAB

function tmp = code(J, K, U)
	t_0 = cos((K / 2.0));
	tmp = ((-2.0 * J) * t_0) * sqrt((1.0 + ((U / ((2.0 * J) * t_0)) ^ 2.0)));
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, N[(N[(N[(-2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * J), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]

Alternative 1: 87.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := -2 \cdot \left|J\right|\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(t\_1 \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;\left(\left(\frac{0.5}{\left|t\_0\right|} \cdot \frac{\left|U\right|}{\left|J\right|}\right) \cdot t\_0\right) \cdot \left(\left|J\right| \cdot -2\right)\\ \mathbf{elif}\;t\_3 \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\left(t\_1 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(\left|J\right| + \left|J\right|\right) \cdot \cos \left(-0.5 \cdot K\right)}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(t\_0 \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{{t\_0}^{2}}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K)))
        (t_1 (* -2.0 (fabs J)))
        (t_2 (cos (/ K 2.0)))
        (t_3
         (*
          (* t_1 t_2)
          (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_2)) 2.0))))))
   (*
    (copysign 1.0 J)
    (if (<= t_3 (- INFINITY))
      (*
       (* (* (/ 0.5 (fabs t_0)) (/ (fabs U) (fabs J))) t_0)
       (* (fabs J) -2.0))
      (if (<= t_3 4e+293)
        (*
         (* t_1 (cos (* K 0.5)))
         (sqrt
          (+ 1.0 (pow (/ U (* (+ (fabs J) (fabs J)) (cos (* -0.5 K)))) 2.0))))
        (* -2.0 (* t_0 (sqrt (* 0.25 (/ (pow U 2.0) (pow t_0 2.0)))))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((0.5 * K));
	double t_1 = -2.0 * fabs(J);
	double t_2 = cos((K / 2.0));
	double t_3 = (t_1 * t_2) * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_2)), 2.0)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = (((0.5 / fabs(t_0)) * (fabs(U) / fabs(J))) * t_0) * (fabs(J) * -2.0);
	} else if (t_3 <= 4e+293) {
		tmp = (t_1 * cos((K * 0.5))) * sqrt((1.0 + pow((U / ((fabs(J) + fabs(J)) * cos((-0.5 * K)))), 2.0)));
	} else {
		tmp = -2.0 * (t_0 * sqrt((0.25 * (pow(U, 2.0) / pow(t_0, 2.0)))));
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((0.5 * K));
	double t_1 = -2.0 * Math.abs(J);
	double t_2 = Math.cos((K / 2.0));
	double t_3 = (t_1 * t_2) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * Math.abs(J)) * t_2)), 2.0)));
	double tmp;
	if (t_3 <= -Double.POSITIVE_INFINITY) {
		tmp = (((0.5 / Math.abs(t_0)) * (Math.abs(U) / Math.abs(J))) * t_0) * (Math.abs(J) * -2.0);
	} else if (t_3 <= 4e+293) {
		tmp = (t_1 * Math.cos((K * 0.5))) * Math.sqrt((1.0 + Math.pow((U / ((Math.abs(J) + Math.abs(J)) * Math.cos((-0.5 * K)))), 2.0)));
	} else {
		tmp = -2.0 * (t_0 * Math.sqrt((0.25 * (Math.pow(U, 2.0) / Math.pow(t_0, 2.0)))));
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((0.5 * K))
	t_1 = -2.0 * math.fabs(J)
	t_2 = math.cos((K / 2.0))
	t_3 = (t_1 * t_2) * math.sqrt((1.0 + math.pow((U / ((2.0 * math.fabs(J)) * t_2)), 2.0)))
	tmp = 0
	if t_3 <= -math.inf:
		tmp = (((0.5 / math.fabs(t_0)) * (math.fabs(U) / math.fabs(J))) * t_0) * (math.fabs(J) * -2.0)
	elif t_3 <= 4e+293:
		tmp = (t_1 * math.cos((K * 0.5))) * math.sqrt((1.0 + math.pow((U / ((math.fabs(J) + math.fabs(J)) * math.cos((-0.5 * K)))), 2.0)))
	else:
		tmp = -2.0 * (t_0 * math.sqrt((0.25 * (math.pow(U, 2.0) / math.pow(t_0, 2.0)))))
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(0.5 * K))
	t_1 = Float64(-2.0 * abs(J))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(t_1 * t_2) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_2)) ^ 2.0))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(0.5 / abs(t_0)) * Float64(abs(U) / abs(J))) * t_0) * Float64(abs(J) * -2.0));
	elseif (t_3 <= 4e+293)
		tmp = Float64(Float64(t_1 * cos(Float64(K * 0.5))) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(abs(J) + abs(J)) * cos(Float64(-0.5 * K)))) ^ 2.0))));
	else
		tmp = Float64(-2.0 * Float64(t_0 * sqrt(Float64(0.25 * Float64((U ^ 2.0) / (t_0 ^ 2.0))))));
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((0.5 * K));
	t_1 = -2.0 * abs(J);
	t_2 = cos((K / 2.0));
	t_3 = (t_1 * t_2) * sqrt((1.0 + ((U / ((2.0 * abs(J)) * t_2)) ^ 2.0)));
	tmp = 0.0;
	if (t_3 <= -Inf)
		tmp = (((0.5 / abs(t_0)) * (abs(U) / abs(J))) * t_0) * (abs(J) * -2.0);
	elseif (t_3 <= 4e+293)
		tmp = (t_1 * cos((K * 0.5))) * sqrt((1.0 + ((U / ((abs(J) + abs(J)) * cos((-0.5 * K)))) ^ 2.0)));
	else
		tmp = -2.0 * (t_0 * sqrt((0.25 * ((U ^ 2.0) / (t_0 ^ 2.0)))));
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], N[(N[(N[(N[(0.5 / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 4e+293], N[(N[(t$95$1 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(t$95$0 * N[Sqrt[N[(0.25 * N[(N[Power[U, 2.0], $MachinePrecision] / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{\color{blue}{J}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      8. lower-*.f64 14.1%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   4. Applied rewrites 14.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J}} \]

   6. Applied rewrites 20.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\left|U\right| \cdot \sqrt{\frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)} \cdot 0.25}}{J} \]

   8. Applied rewrites 21.0%

      \[\leadsto \color{blue}{\left(\left(\frac{0.5}{\left|\cos \left(0.5 \cdot K\right)\right|} \cdot \frac{\left|U\right|}{J}\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right)} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 3.9999999999999997e293

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(\frac{K}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. mult-flip N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      4. lower-*.f64 73.8%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Applied rewrites 73.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \color{blue}{\left(\frac{K}{2}\right)}}\right)}^{2}} \]

      2. mult-flip N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}}\right)}^{2}} \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(K \cdot \color{blue}{\frac{1}{2}}\right)}\right)}^{2}} \]

      4. lower-*.f64 73.8%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}}\right)}^{2}} \]

   5. Applied rewrites 73.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}}\right)}^{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{1 + \color{blue}{{\left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right)}^{2}}} \]

   if 3.9999999999999997e293 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \color{blue}{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\color{blue}{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\color{blue}{\frac{1}{4}} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      9. lower-pow.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      10. lower-cos.f64 N/A

          \[\leadsto -2 \cdot \left(\cos \left(\frac{1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right) \]

      11. lower-*.f64 15.5%

          \[\leadsto -2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}\right) \]

   4. Applied rewrites 15.5%

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 87.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \cos \left(-0.5 \cdot K\right)\\ t_2 := -2 \cdot \left|J\right|\\ t_3 := \cos \left(\frac{K}{2}\right)\\ t_4 := \left(t\_2 \cdot t\_3\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_3}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\left(\left(\frac{0.5}{\left|t\_0\right|} \cdot \frac{\left|U\right|}{\left|J\right|}\right) \cdot t\_0\right) \cdot \left(\left|J\right| \cdot -2\right)\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\left(t\_2 \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(\left|J\right| + \left|J\right|\right) \cdot t\_1}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(t\_1 \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{0.5 + 0.5 \cdot \cos K}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K)))
        (t_1 (cos (* -0.5 K)))
        (t_2 (* -2.0 (fabs J)))
        (t_3 (cos (/ K 2.0)))
        (t_4
         (*
          (* t_2 t_3)
          (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_3)) 2.0))))))
   (*
    (copysign 1.0 J)
    (if (<= t_4 (- INFINITY))
      (*
       (* (* (/ 0.5 (fabs t_0)) (/ (fabs U) (fabs J))) t_0)
       (* (fabs J) -2.0))
      (if (<= t_4 4e+293)
        (*
         (* t_2 (cos (* K 0.5)))
         (sqrt (+ 1.0 (pow (/ U (* (+ (fabs J) (fabs J)) t_1)) 2.0))))
        (*
         -2.0
         (* t_1 (sqrt (* 0.25 (/ (pow U 2.0) (+ 0.5 (* 0.5 (cos K)))))))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((0.5 * K));
	double t_1 = cos((-0.5 * K));
	double t_2 = -2.0 * fabs(J);
	double t_3 = cos((K / 2.0));
	double t_4 = (t_2 * t_3) * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_3)), 2.0)));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = (((0.5 / fabs(t_0)) * (fabs(U) / fabs(J))) * t_0) * (fabs(J) * -2.0);
	} else if (t_4 <= 4e+293) {
		tmp = (t_2 * cos((K * 0.5))) * sqrt((1.0 + pow((U / ((fabs(J) + fabs(J)) * t_1)), 2.0)));
	} else {
		tmp = -2.0 * (t_1 * sqrt((0.25 * (pow(U, 2.0) / (0.5 + (0.5 * cos(K)))))));
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = Math.cos((0.5 * K));
	double t_1 = Math.cos((-0.5 * K));
	double t_2 = -2.0 * Math.abs(J);
	double t_3 = Math.cos((K / 2.0));
	double t_4 = (t_2 * t_3) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * Math.abs(J)) * t_3)), 2.0)));
	double tmp;
	if (t_4 <= -Double.POSITIVE_INFINITY) {
		tmp = (((0.5 / Math.abs(t_0)) * (Math.abs(U) / Math.abs(J))) * t_0) * (Math.abs(J) * -2.0);
	} else if (t_4 <= 4e+293) {
		tmp = (t_2 * Math.cos((K * 0.5))) * Math.sqrt((1.0 + Math.pow((U / ((Math.abs(J) + Math.abs(J)) * t_1)), 2.0)));
	} else {
		tmp = -2.0 * (t_1 * Math.sqrt((0.25 * (Math.pow(U, 2.0) / (0.5 + (0.5 * Math.cos(K)))))));
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = math.cos((0.5 * K))
	t_1 = math.cos((-0.5 * K))
	t_2 = -2.0 * math.fabs(J)
	t_3 = math.cos((K / 2.0))
	t_4 = (t_2 * t_3) * math.sqrt((1.0 + math.pow((U / ((2.0 * math.fabs(J)) * t_3)), 2.0)))
	tmp = 0
	if t_4 <= -math.inf:
		tmp = (((0.5 / math.fabs(t_0)) * (math.fabs(U) / math.fabs(J))) * t_0) * (math.fabs(J) * -2.0)
	elif t_4 <= 4e+293:
		tmp = (t_2 * math.cos((K * 0.5))) * math.sqrt((1.0 + math.pow((U / ((math.fabs(J) + math.fabs(J)) * t_1)), 2.0)))
	else:
		tmp = -2.0 * (t_1 * math.sqrt((0.25 * (math.pow(U, 2.0) / (0.5 + (0.5 * math.cos(K)))))))
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = cos(Float64(0.5 * K))
	t_1 = cos(Float64(-0.5 * K))
	t_2 = Float64(-2.0 * abs(J))
	t_3 = cos(Float64(K / 2.0))
	t_4 = Float64(Float64(t_2 * t_3) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_3)) ^ 2.0))))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(0.5 / abs(t_0)) * Float64(abs(U) / abs(J))) * t_0) * Float64(abs(J) * -2.0));
	elseif (t_4 <= 4e+293)
		tmp = Float64(Float64(t_2 * cos(Float64(K * 0.5))) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(abs(J) + abs(J)) * t_1)) ^ 2.0))));
	else
		tmp = Float64(-2.0 * Float64(t_1 * sqrt(Float64(0.25 * Float64((U ^ 2.0) / Float64(0.5 + Float64(0.5 * cos(K))))))));
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = cos((0.5 * K));
	t_1 = cos((-0.5 * K));
	t_2 = -2.0 * abs(J);
	t_3 = cos((K / 2.0));
	t_4 = (t_2 * t_3) * sqrt((1.0 + ((U / ((2.0 * abs(J)) * t_3)) ^ 2.0)));
	tmp = 0.0;
	if (t_4 <= -Inf)
		tmp = (((0.5 / abs(t_0)) * (abs(U) / abs(J))) * t_0) * (abs(J) * -2.0);
	elseif (t_4 <= 4e+293)
		tmp = (t_2 * cos((K * 0.5))) * sqrt((1.0 + ((U / ((abs(J) + abs(J)) * t_1)) ^ 2.0)));
	else
		tmp = -2.0 * (t_1 * sqrt((0.25 * ((U ^ 2.0) / (0.5 + (0.5 * cos(K)))))));
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$2 * t$95$3), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[(N[(0.5 / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 4e+293], N[(N[(t$95$2 * N[Cos[N[(K * 0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(-2.0 * N[(t$95$1 * N[Sqrt[N[(0.25 * N[(N[Power[U, 2.0], $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{\color{blue}{J}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      8. lower-*.f64 14.1%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   4. Applied rewrites 14.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J}} \]

   6. Applied rewrites 20.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\left|U\right| \cdot \sqrt{\frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)} \cdot 0.25}}{J} \]

   8. Applied rewrites 21.0%

      \[\leadsto \color{blue}{\left(\left(\frac{0.5}{\left|\cos \left(0.5 \cdot K\right)\right|} \cdot \frac{\left|U\right|}{J}\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right)} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 3.9999999999999997e293

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(\frac{K}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      2. mult-flip N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot \color{blue}{\frac{1}{2}}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

      4. lower-*.f64 73.8%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Applied rewrites 73.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \color{blue}{\left(\frac{K}{2}\right)}}\right)}^{2}} \]

      2. mult-flip N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{2}\right)}}\right)}^{2}} \]

      3. metadata-eval N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot \frac{1}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(K \cdot \color{blue}{\frac{1}{2}}\right)}\right)}^{2}} \]

      4. lower-*.f64 73.8%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}}\right)}^{2}} \]

   5. Applied rewrites 73.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \color{blue}{\left(K \cdot 0.5\right)}}\right)}^{2}} \]
   6. Step-by-step derivation

   7. Applied rewrites 73.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(K \cdot 0.5\right)\right) \cdot \sqrt{1 + \color{blue}{{\left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right)}^{2}}} \]

   if 3.9999999999999997e293 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{0.5 + 0.5 \cdot \cos K} - -1}} \]

   4. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \color{blue}{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \sqrt{\color{blue}{\frac{1}{4} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \sqrt{\color{blue}{\frac{1}{4}} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto -2 \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}\right) \]

      11. lower-cos.f64 15.4%

          \[\leadsto -2 \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{0.5 + 0.5 \cdot \cos K}}\right) \]

   6. Applied rewrites 15.4%

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{0.5 + 0.5 \cdot \cos K}}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{U}{\left|J\right|}\\ t_1 := \cos \left(0.5 \cdot K\right)\\ t_2 := 0.5 + 0.5 \cdot \cos K\\ t_3 := \left|J\right| \cdot -2\\ t_4 := \cos \left(-0.5 \cdot K\right)\\ t_5 := \cos \left(\frac{K}{2}\right)\\ t_6 := \left(\left(-2 \cdot \left|J\right|\right) \cdot t\_5\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_5}\right)}^{2}}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_6 \leq -\infty:\\ \;\;\;\;\left(\left(\frac{0.5}{\left|t\_1\right|} \cdot \frac{\left|U\right|}{\left|J\right|}\right) \cdot t\_1\right) \cdot t\_3\\ \mathbf{elif}\;t\_6 \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{t\_0 \cdot t\_0}{4}}{t\_2} - -1} \cdot t\_4\right) \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;-2 \cdot \left(t\_4 \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{t\_2}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (/ U (fabs J)))
        (t_1 (cos (* 0.5 K)))
        (t_2 (+ 0.5 (* 0.5 (cos K))))
        (t_3 (* (fabs J) -2.0))
        (t_4 (cos (* -0.5 K)))
        (t_5 (cos (/ K 2.0)))
        (t_6
         (*
          (* (* -2.0 (fabs J)) t_5)
          (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_5)) 2.0))))))
   (*
    (copysign 1.0 J)
    (if (<= t_6 (- INFINITY))
      (* (* (* (/ 0.5 (fabs t_1)) (/ (fabs U) (fabs J))) t_1) t_3)
      (if (<= t_6 4e+293)
        (* (* (sqrt (- (/ (/ (* t_0 t_0) 4.0) t_2) -1.0)) t_4) t_3)
        (* -2.0 (* t_4 (sqrt (* 0.25 (/ (pow U 2.0) t_2))))))))))

C

double code(double J, double K, double U) {
	double t_0 = U / fabs(J);
	double t_1 = cos((0.5 * K));
	double t_2 = 0.5 + (0.5 * cos(K));
	double t_3 = fabs(J) * -2.0;
	double t_4 = cos((-0.5 * K));
	double t_5 = cos((K / 2.0));
	double t_6 = ((-2.0 * fabs(J)) * t_5) * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_5)), 2.0)));
	double tmp;
	if (t_6 <= -((double) INFINITY)) {
		tmp = (((0.5 / fabs(t_1)) * (fabs(U) / fabs(J))) * t_1) * t_3;
	} else if (t_6 <= 4e+293) {
		tmp = (sqrt(((((t_0 * t_0) / 4.0) / t_2) - -1.0)) * t_4) * t_3;
	} else {
		tmp = -2.0 * (t_4 * sqrt((0.25 * (pow(U, 2.0) / t_2))));
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = U / Math.abs(J);
	double t_1 = Math.cos((0.5 * K));
	double t_2 = 0.5 + (0.5 * Math.cos(K));
	double t_3 = Math.abs(J) * -2.0;
	double t_4 = Math.cos((-0.5 * K));
	double t_5 = Math.cos((K / 2.0));
	double t_6 = ((-2.0 * Math.abs(J)) * t_5) * Math.sqrt((1.0 + Math.pow((U / ((2.0 * Math.abs(J)) * t_5)), 2.0)));
	double tmp;
	if (t_6 <= -Double.POSITIVE_INFINITY) {
		tmp = (((0.5 / Math.abs(t_1)) * (Math.abs(U) / Math.abs(J))) * t_1) * t_3;
	} else if (t_6 <= 4e+293) {
		tmp = (Math.sqrt(((((t_0 * t_0) / 4.0) / t_2) - -1.0)) * t_4) * t_3;
	} else {
		tmp = -2.0 * (t_4 * Math.sqrt((0.25 * (Math.pow(U, 2.0) / t_2))));
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = U / math.fabs(J)
	t_1 = math.cos((0.5 * K))
	t_2 = 0.5 + (0.5 * math.cos(K))
	t_3 = math.fabs(J) * -2.0
	t_4 = math.cos((-0.5 * K))
	t_5 = math.cos((K / 2.0))
	t_6 = ((-2.0 * math.fabs(J)) * t_5) * math.sqrt((1.0 + math.pow((U / ((2.0 * math.fabs(J)) * t_5)), 2.0)))
	tmp = 0
	if t_6 <= -math.inf:
		tmp = (((0.5 / math.fabs(t_1)) * (math.fabs(U) / math.fabs(J))) * t_1) * t_3
	elif t_6 <= 4e+293:
		tmp = (math.sqrt(((((t_0 * t_0) / 4.0) / t_2) - -1.0)) * t_4) * t_3
	else:
		tmp = -2.0 * (t_4 * math.sqrt((0.25 * (math.pow(U, 2.0) / t_2))))
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = Float64(U / abs(J))
	t_1 = cos(Float64(0.5 * K))
	t_2 = Float64(0.5 + Float64(0.5 * cos(K)))
	t_3 = Float64(abs(J) * -2.0)
	t_4 = cos(Float64(-0.5 * K))
	t_5 = cos(Float64(K / 2.0))
	t_6 = Float64(Float64(Float64(-2.0 * abs(J)) * t_5) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_5)) ^ 2.0))))
	tmp = 0.0
	if (t_6 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(Float64(0.5 / abs(t_1)) * Float64(abs(U) / abs(J))) * t_1) * t_3);
	elseif (t_6 <= 4e+293)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(Float64(t_0 * t_0) / 4.0) / t_2) - -1.0)) * t_4) * t_3);
	else
		tmp = Float64(-2.0 * Float64(t_4 * sqrt(Float64(0.25 * Float64((U ^ 2.0) / t_2)))));
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = U / abs(J);
	t_1 = cos((0.5 * K));
	t_2 = 0.5 + (0.5 * cos(K));
	t_3 = abs(J) * -2.0;
	t_4 = cos((-0.5 * K));
	t_5 = cos((K / 2.0));
	t_6 = ((-2.0 * abs(J)) * t_5) * sqrt((1.0 + ((U / ((2.0 * abs(J)) * t_5)) ^ 2.0)));
	tmp = 0.0;
	if (t_6 <= -Inf)
		tmp = (((0.5 / abs(t_1)) * (abs(U) / abs(J))) * t_1) * t_3;
	elseif (t_6 <= 4e+293)
		tmp = (sqrt(((((t_0 * t_0) / 4.0) / t_2) - -1.0)) * t_4) * t_3;
	else
		tmp = -2.0 * (t_4 * sqrt((0.25 * ((U ^ 2.0) / t_2))));
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(U / N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$6 = N[(N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$5), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$6, (-Infinity)], N[(N[(N[(N[(0.5 / N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$3), $MachinePrecision], If[LessEqual[t$95$6, 4e+293], N[(N[(N[Sqrt[N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] / t$95$2), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * t$95$4), $MachinePrecision] * t$95$3), $MachinePrecision], N[(-2.0 * N[(t$95$4 * N[Sqrt[N[(0.25 * N[(N[Power[U, 2.0], $MachinePrecision] / t$95$2), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{\color{blue}{J}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      8. lower-*.f64 14.1%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   4. Applied rewrites 14.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J}} \]

   6. Applied rewrites 20.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\left|U\right| \cdot \sqrt{\frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)} \cdot 0.25}}{J} \]

   8. Applied rewrites 21.0%

      \[\leadsto \color{blue}{\left(\left(\frac{0.5}{\left|\cos \left(0.5 \cdot K\right)\right|} \cdot \frac{\left|U\right|}{J}\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right)} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 3.9999999999999997e293

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{0.5 + 0.5 \cdot \cos K} - -1} \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right)} \]

   if 3.9999999999999997e293 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{0.5 + 0.5 \cdot \cos K} - -1}} \]

   4. Taylor expanded in J around 0

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto -2 \cdot \color{blue}{\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}\right)} \]

      2. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \color{blue}{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}}\right) \]

      3. lower-cos.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \sqrt{\color{blue}{\frac{1}{4} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \sqrt{\color{blue}{\frac{1}{4}} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}\right) \]

      5. lower-sqrt.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}\right) \]

      8. lower-pow.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}\right) \]

      9. lower-+.f64 N/A

         \[\leadsto -2 \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}\right) \]

      10. lower-*.f64 N/A

          \[\leadsto -2 \cdot \left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot \sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}}\right) \]

      11. lower-cos.f64 15.4%

          \[\leadsto -2 \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{0.5 + 0.5 \cdot \cos K}}\right) \]

   6. Applied rewrites 15.4%

      \[\leadsto \color{blue}{-2 \cdot \left(\cos \left(-0.5 \cdot K\right) \cdot \sqrt{0.25 \cdot \frac{{U}^{2}}{0.5 + 0.5 \cdot \cos K}}\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 87.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{U}{\left|J\right|}\\ t_1 := \cos \left(0.5 \cdot K\right)\\ t_2 := \left|J\right| \cdot -2\\ t_3 := \cos \left(\frac{K}{2}\right)\\ t_4 := \left(-2 \cdot \left|J\right|\right) \cdot t\_3\\ t_5 := t\_4 \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_3}\right)}^{2}}\\ t_6 := \frac{0.5}{\left|t\_1\right|}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;\left(\left(t\_6 \cdot \frac{\left|U\right|}{\left|J\right|}\right) \cdot t\_1\right) \cdot t\_2\\ \mathbf{elif}\;t\_5 \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\left(\sqrt{\frac{\frac{t\_0 \cdot t\_0}{4}}{0.5 + 0.5 \cdot \cos K} - -1} \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_4 \cdot \frac{\left|t\_6 \cdot U\right|}{\left|J\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (/ U (fabs J)))
        (t_1 (cos (* 0.5 K)))
        (t_2 (* (fabs J) -2.0))
        (t_3 (cos (/ K 2.0)))
        (t_4 (* (* -2.0 (fabs J)) t_3))
        (t_5 (* t_4 (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_3)) 2.0)))))
        (t_6 (/ 0.5 (fabs t_1))))
   (*
    (copysign 1.0 J)
    (if (<= t_5 (- INFINITY))
      (* (* (* t_6 (/ (fabs U) (fabs J))) t_1) t_2)
      (if (<= t_5 4e+293)
        (*
         (*
          (sqrt (- (/ (/ (* t_0 t_0) 4.0) (+ 0.5 (* 0.5 (cos K)))) -1.0))
          (cos (* -0.5 K)))
         t_2)
        (* t_4 (/ (fabs (* t_6 U)) (fabs J))))))))

C

double code(double J, double K, double U) {
	double t_0 = U / fabs(J);
	double t_1 = cos((0.5 * K));
	double t_2 = fabs(J) * -2.0;
	double t_3 = cos((K / 2.0));
	double t_4 = (-2.0 * fabs(J)) * t_3;
	double t_5 = t_4 * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_3)), 2.0)));
	double t_6 = 0.5 / fabs(t_1);
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = ((t_6 * (fabs(U) / fabs(J))) * t_1) * t_2;
	} else if (t_5 <= 4e+293) {
		tmp = (sqrt(((((t_0 * t_0) / 4.0) / (0.5 + (0.5 * cos(K)))) - -1.0)) * cos((-0.5 * K))) * t_2;
	} else {
		tmp = t_4 * (fabs((t_6 * U)) / fabs(J));
	}
	return copysign(1.0, J) * tmp;
}

Java

public static double code(double J, double K, double U) {
	double t_0 = U / Math.abs(J);
	double t_1 = Math.cos((0.5 * K));
	double t_2 = Math.abs(J) * -2.0;
	double t_3 = Math.cos((K / 2.0));
	double t_4 = (-2.0 * Math.abs(J)) * t_3;
	double t_5 = t_4 * Math.sqrt((1.0 + Math.pow((U / ((2.0 * Math.abs(J)) * t_3)), 2.0)));
	double t_6 = 0.5 / Math.abs(t_1);
	double tmp;
	if (t_5 <= -Double.POSITIVE_INFINITY) {
		tmp = ((t_6 * (Math.abs(U) / Math.abs(J))) * t_1) * t_2;
	} else if (t_5 <= 4e+293) {
		tmp = (Math.sqrt(((((t_0 * t_0) / 4.0) / (0.5 + (0.5 * Math.cos(K)))) - -1.0)) * Math.cos((-0.5 * K))) * t_2;
	} else {
		tmp = t_4 * (Math.abs((t_6 * U)) / Math.abs(J));
	}
	return Math.copySign(1.0, J) * tmp;
}

Python

def code(J, K, U):
	t_0 = U / math.fabs(J)
	t_1 = math.cos((0.5 * K))
	t_2 = math.fabs(J) * -2.0
	t_3 = math.cos((K / 2.0))
	t_4 = (-2.0 * math.fabs(J)) * t_3
	t_5 = t_4 * math.sqrt((1.0 + math.pow((U / ((2.0 * math.fabs(J)) * t_3)), 2.0)))
	t_6 = 0.5 / math.fabs(t_1)
	tmp = 0
	if t_5 <= -math.inf:
		tmp = ((t_6 * (math.fabs(U) / math.fabs(J))) * t_1) * t_2
	elif t_5 <= 4e+293:
		tmp = (math.sqrt(((((t_0 * t_0) / 4.0) / (0.5 + (0.5 * math.cos(K)))) - -1.0)) * math.cos((-0.5 * K))) * t_2
	else:
		tmp = t_4 * (math.fabs((t_6 * U)) / math.fabs(J))
	return math.copysign(1.0, J) * tmp

Julia

function code(J, K, U)
	t_0 = Float64(U / abs(J))
	t_1 = cos(Float64(0.5 * K))
	t_2 = Float64(abs(J) * -2.0)
	t_3 = cos(Float64(K / 2.0))
	t_4 = Float64(Float64(-2.0 * abs(J)) * t_3)
	t_5 = Float64(t_4 * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_3)) ^ 2.0))))
	t_6 = Float64(0.5 / abs(t_1))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t_6 * Float64(abs(U) / abs(J))) * t_1) * t_2);
	elseif (t_5 <= 4e+293)
		tmp = Float64(Float64(sqrt(Float64(Float64(Float64(Float64(t_0 * t_0) / 4.0) / Float64(0.5 + Float64(0.5 * cos(K)))) - -1.0)) * cos(Float64(-0.5 * K))) * t_2);
	else
		tmp = Float64(t_4 * Float64(abs(Float64(t_6 * U)) / abs(J)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

MATLAB

function tmp_2 = code(J, K, U)
	t_0 = U / abs(J);
	t_1 = cos((0.5 * K));
	t_2 = abs(J) * -2.0;
	t_3 = cos((K / 2.0));
	t_4 = (-2.0 * abs(J)) * t_3;
	t_5 = t_4 * sqrt((1.0 + ((U / ((2.0 * abs(J)) * t_3)) ^ 2.0)));
	t_6 = 0.5 / abs(t_1);
	tmp = 0.0;
	if (t_5 <= -Inf)
		tmp = ((t_6 * (abs(U) / abs(J))) * t_1) * t_2;
	elseif (t_5 <= 4e+293)
		tmp = (sqrt(((((t_0 * t_0) / 4.0) / (0.5 + (0.5 * cos(K)))) - -1.0)) * cos((-0.5 * K))) * t_2;
	else
		tmp = t_4 * (abs((t_6 * U)) / abs(J));
	end
	tmp_2 = (sign(J) * abs(1.0)) * tmp;
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(U / N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(0.5 / N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$5, (-Infinity)], N[(N[(N[(t$95$6 * N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$5, 4e+293], N[(N[(N[Sqrt[N[(N[(N[(N[(t$95$0 * t$95$0), $MachinePrecision] / 4.0), $MachinePrecision] / N[(0.5 + N[(0.5 * N[Cos[K], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(t$95$4 * N[(N[Abs[N[(t$95$6 * U), $MachinePrecision]], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{\color{blue}{J}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      8. lower-*.f64 14.1%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   4. Applied rewrites 14.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J}} \]

   6. Applied rewrites 20.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\left|U\right| \cdot \sqrt{\frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)} \cdot 0.25}}{J} \]

   8. Applied rewrites 21.0%

      \[\leadsto \color{blue}{\left(\left(\frac{0.5}{\left|\cos \left(0.5 \cdot K\right)\right|} \cdot \frac{\left|U\right|}{J}\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right)} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 3.9999999999999997e293

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\left(\sqrt{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{0.5 + 0.5 \cdot \cos K} - -1} \cdot \cos \left(-0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right)} \]

   if 3.9999999999999997e293 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{\color{blue}{J}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      8. lower-*.f64 14.1%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   4. Applied rewrites 14.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J}} \]

   6. Applied rewrites 20.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\left|U\right| \cdot \sqrt{\frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)} \cdot 0.25}}{J} \]

   8. Applied rewrites 20.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\left|\frac{0.5}{\left|\cos \left(0.5 \cdot K\right)\right|} \cdot U\right|}{\color{blue}{J}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 86.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := \frac{U}{\left|J\right|}\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(-2 \cdot \left|J\right|\right) \cdot t\_2\\ t_4 := t\_3 \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ t_5 := \frac{0.5}{\left|t\_0\right|}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\left(\left(t\_5 \cdot \frac{\left|U\right|}{\left|J\right|}\right) \cdot t\_0\right) \cdot \left(\left|J\right| \cdot -2\right)\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot \left|J\right|\right) \cdot \sqrt{\mathsf{fma}\left(t\_1, \frac{t\_1}{\cos K - -1} \cdot 0.5, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \frac{\left|t\_5 \cdot U\right|}{\left|J\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K)))
        (t_1 (/ U (fabs J)))
        (t_2 (cos (/ K 2.0)))
        (t_3 (* (* -2.0 (fabs J)) t_2))
        (t_4 (* t_3 (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_2)) 2.0)))))
        (t_5 (/ 0.5 (fabs t_0))))
   (*
    (copysign 1.0 J)
    (if (<= t_4 (- INFINITY))
      (* (* (* t_5 (/ (fabs U) (fabs J))) t_0) (* (fabs J) -2.0))
      (if (<= t_4 4e+293)
        (*
         (* (* (cos (* -0.5 K)) -2.0) (fabs J))
         (sqrt (fma t_1 (* (/ t_1 (- (cos K) -1.0)) 0.5) 1.0)))
        (* t_3 (/ (fabs (* t_5 U)) (fabs J))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((0.5 * K));
	double t_1 = U / fabs(J);
	double t_2 = cos((K / 2.0));
	double t_3 = (-2.0 * fabs(J)) * t_2;
	double t_4 = t_3 * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_2)), 2.0)));
	double t_5 = 0.5 / fabs(t_0);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = ((t_5 * (fabs(U) / fabs(J))) * t_0) * (fabs(J) * -2.0);
	} else if (t_4 <= 4e+293) {
		tmp = ((cos((-0.5 * K)) * -2.0) * fabs(J)) * sqrt(fma(t_1, ((t_1 / (cos(K) - -1.0)) * 0.5), 1.0));
	} else {
		tmp = t_3 * (fabs((t_5 * U)) / fabs(J));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(0.5 * K))
	t_1 = Float64(U / abs(J))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(-2.0 * abs(J)) * t_2)
	t_4 = Float64(t_3 * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_2)) ^ 2.0))))
	t_5 = Float64(0.5 / abs(t_0))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t_5 * Float64(abs(U) / abs(J))) * t_0) * Float64(abs(J) * -2.0));
	elseif (t_4 <= 4e+293)
		tmp = Float64(Float64(Float64(cos(Float64(-0.5 * K)) * -2.0) * abs(J)) * sqrt(fma(t_1, Float64(Float64(t_1 / Float64(cos(K) - -1.0)) * 0.5), 1.0)));
	else
		tmp = Float64(t_3 * Float64(abs(Float64(t_5 * U)) / abs(J)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(U / N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(0.5 / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[(t$95$5 * N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 4e+293], N[(N[(N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * -2.0), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[Sqrt[N[(t$95$1 * N[(N[(t$95$1 / N[(N[Cos[K], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(t$95$3 * N[(N[Abs[N[(t$95$5 * U), $MachinePrecision]], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{\color{blue}{J}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      8. lower-*.f64 14.1%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   4. Applied rewrites 14.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J}} \]

   6. Applied rewrites 20.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\left|U\right| \cdot \sqrt{\frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)} \cdot 0.25}}{J} \]

   8. Applied rewrites 21.0%

      \[\leadsto \color{blue}{\left(\left(\frac{0.5}{\left|\cos \left(0.5 \cdot K\right)\right|} \cdot \frac{\left|U\right|}{J}\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right)} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 3.9999999999999997e293

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{0.5 + 0.5 \cdot \cos K} - -1}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\color{blue}{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{\color{blue}{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K} - -1} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{\frac{\color{blue}{\frac{U}{J} \cdot \frac{U}{J}}}{4}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K} - -1} \]

      4. associate-/l* N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{\color{blue}{\frac{U}{J} \cdot \frac{\frac{U}{J}}{4}}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K} - -1} \]

      5. associate-/l* N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\color{blue}{\frac{U}{J} \cdot \frac{\frac{\frac{U}{J}}{4}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}} - -1} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\color{blue}{\frac{U}{J}} \cdot \frac{\frac{\frac{U}{J}}{4}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K} - -1} \]

      7. frac-times N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\color{blue}{\frac{U \cdot \frac{\frac{U}{J}}{4}}{J \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos K\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\color{blue}{\frac{U \cdot \frac{\frac{U}{J}}{4}}{J \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos K\right)}} - -1} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{\color{blue}{U \cdot \frac{\frac{U}{J}}{4}}}{J \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos K\right)} - -1} \]

      10. mult-flip N/A

          \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U \cdot \color{blue}{\left(\frac{U}{J} \cdot \frac{1}{4}\right)}}{J \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos K\right)} - -1} \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U \cdot \left(\frac{U}{J} \cdot \color{blue}{\frac{1}{4}}\right)}{J \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos K\right)} - -1} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U \cdot \color{blue}{\left(\frac{U}{J} \cdot \frac{1}{4}\right)}}{J \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos K\right)} - -1} \]

      13. lower-*.f64 70.8%

          \[\leadsto \left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U \cdot \left(\frac{U}{J} \cdot 0.25\right)}{\color{blue}{J \cdot \left(0.5 + 0.5 \cdot \cos K\right)}} - -1} \]

      14. lift-+.f64 N/A

          \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U \cdot \left(\frac{U}{J} \cdot \frac{1}{4}\right)}{J \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos K\right)}} - -1} \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U \cdot \left(\frac{U}{J} \cdot \frac{1}{4}\right)}{J \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos K + \frac{1}{2}\right)}} - -1} \]

      16. lift-*.f64 N/A

          \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U \cdot \left(\frac{U}{J} \cdot \frac{1}{4}\right)}{J \cdot \left(\color{blue}{\frac{1}{2} \cdot \cos K} + \frac{1}{2}\right)} - -1} \]

      17. *-commutative N/A

          \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U \cdot \left(\frac{U}{J} \cdot \frac{1}{4}\right)}{J \cdot \left(\color{blue}{\cos K \cdot \frac{1}{2}} + \frac{1}{2}\right)} - -1} \]

      18. lower-fma.f64 70.8%

          \[\leadsto \left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U \cdot \left(\frac{U}{J} \cdot 0.25\right)}{J \cdot \color{blue}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right)}} - -1} \]

   5. Applied rewrites 70.8%

      \[\leadsto \left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\color{blue}{\frac{U \cdot \left(\frac{U}{J} \cdot 0.25\right)}{J \cdot \mathsf{fma}\left(\cos K, 0.5, 0.5\right)}} - -1} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\color{blue}{\frac{U \cdot \left(\frac{U}{J} \cdot \frac{1}{4}\right)}{J \cdot \mathsf{fma}\left(\cos K, \frac{1}{2}, \frac{1}{2}\right)} - -1}} \]

      2. sub-flip N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\color{blue}{\frac{U \cdot \left(\frac{U}{J} \cdot \frac{1}{4}\right)}{J \cdot \mathsf{fma}\left(\cos K, \frac{1}{2}, \frac{1}{2}\right)} + \left(\mathsf{neg}\left(-1\right)\right)}} \]

      3. lift-/.f64 N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\color{blue}{\frac{U \cdot \left(\frac{U}{J} \cdot \frac{1}{4}\right)}{J \cdot \mathsf{fma}\left(\cos K, \frac{1}{2}, \frac{1}{2}\right)}} + \left(\mathsf{neg}\left(-1\right)\right)} \]

      4. lift-*.f64 N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{\color{blue}{U \cdot \left(\frac{U}{J} \cdot \frac{1}{4}\right)}}{J \cdot \mathsf{fma}\left(\cos K, \frac{1}{2}, \frac{1}{2}\right)} + \left(\mathsf{neg}\left(-1\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U \cdot \left(\frac{U}{J} \cdot \frac{1}{4}\right)}{\color{blue}{J \cdot \mathsf{fma}\left(\cos K, \frac{1}{2}, \frac{1}{2}\right)}} + \left(\mathsf{neg}\left(-1\right)\right)} \]

      6. times-frac N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\color{blue}{\frac{U}{J} \cdot \frac{\frac{U}{J} \cdot \frac{1}{4}}{\mathsf{fma}\left(\cos K, \frac{1}{2}, \frac{1}{2}\right)}} + \left(\mathsf{neg}\left(-1\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\color{blue}{\frac{U}{J}} \cdot \frac{\frac{U}{J} \cdot \frac{1}{4}}{\mathsf{fma}\left(\cos K, \frac{1}{2}, \frac{1}{2}\right)} + \left(\mathsf{neg}\left(-1\right)\right)} \]

      8. metadata-eval N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U}{J} \cdot \frac{\frac{U}{J} \cdot \frac{1}{4}}{\mathsf{fma}\left(\cos K, \frac{1}{2}, \frac{1}{2}\right)} + \color{blue}{1}} \]

      9. lower-fma.f64 N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J}, \frac{\frac{U}{J} \cdot \frac{1}{4}}{\mathsf{fma}\left(\cos K, \frac{1}{2}, \frac{1}{2}\right)}, 1\right)}} \]

   7. Applied rewrites 73.7%

      \[\leadsto \left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\color{blue}{\mathsf{fma}\left(\frac{U}{J}, \frac{\frac{U}{J}}{\cos K - -1} \cdot 0.5, 1\right)}} \]

   if 3.9999999999999997e293 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{\color{blue}{J}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      8. lower-*.f64 14.1%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   4. Applied rewrites 14.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J}} \]

   6. Applied rewrites 20.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\left|U\right| \cdot \sqrt{\frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)} \cdot 0.25}}{J} \]

   8. Applied rewrites 20.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\left|\frac{0.5}{\left|\cos \left(0.5 \cdot K\right)\right|} \cdot U\right|}{\color{blue}{J}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 84.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \cos \left(0.5 \cdot K\right)\\ t_1 := -2 \cdot \left|J\right|\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := t\_1 \cdot t\_2\\ t_4 := t\_3 \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ t_5 := \frac{0.5}{\left|t\_0\right|}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;\left(\left(t\_5 \cdot \frac{\left|U\right|}{\left|J\right|}\right) \cdot t\_0\right) \cdot \left(\left|J\right| \cdot -2\right)\\ \mathbf{elif}\;t\_4 \leq 4 \cdot 10^{+293}:\\ \;\;\;\;\left(\sqrt{\mathsf{fma}\left(\frac{\frac{\frac{U}{\left|J\right|}}{\cos K - -1} \cdot 0.5}{\left|J\right|}, U, 1\right)} \cdot \cos \left(K \cdot -0.5\right)\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3 \cdot \frac{\left|t\_5 \cdot U\right|}{\left|J\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (cos (* 0.5 K)))
        (t_1 (* -2.0 (fabs J)))
        (t_2 (cos (/ K 2.0)))
        (t_3 (* t_1 t_2))
        (t_4 (* t_3 (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_2)) 2.0)))))
        (t_5 (/ 0.5 (fabs t_0))))
   (*
    (copysign 1.0 J)
    (if (<= t_4 (- INFINITY))
      (* (* (* t_5 (/ (fabs U) (fabs J))) t_0) (* (fabs J) -2.0))
      (if (<= t_4 4e+293)
        (*
         (*
          (sqrt
           (fma
            (/ (* (/ (/ U (fabs J)) (- (cos K) -1.0)) 0.5) (fabs J))
            U
            1.0))
          (cos (* K -0.5)))
         t_1)
        (* t_3 (/ (fabs (* t_5 U)) (fabs J))))))))

C

double code(double J, double K, double U) {
	double t_0 = cos((0.5 * K));
	double t_1 = -2.0 * fabs(J);
	double t_2 = cos((K / 2.0));
	double t_3 = t_1 * t_2;
	double t_4 = t_3 * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_2)), 2.0)));
	double t_5 = 0.5 / fabs(t_0);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = ((t_5 * (fabs(U) / fabs(J))) * t_0) * (fabs(J) * -2.0);
	} else if (t_4 <= 4e+293) {
		tmp = (sqrt(fma(((((U / fabs(J)) / (cos(K) - -1.0)) * 0.5) / fabs(J)), U, 1.0)) * cos((K * -0.5))) * t_1;
	} else {
		tmp = t_3 * (fabs((t_5 * U)) / fabs(J));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = cos(Float64(0.5 * K))
	t_1 = Float64(-2.0 * abs(J))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(t_1 * t_2)
	t_4 = Float64(t_3 * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_2)) ^ 2.0))))
	t_5 = Float64(0.5 / abs(t_0))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t_5 * Float64(abs(U) / abs(J))) * t_0) * Float64(abs(J) * -2.0));
	elseif (t_4 <= 4e+293)
		tmp = Float64(Float64(sqrt(fma(Float64(Float64(Float64(Float64(U / abs(J)) / Float64(cos(K) - -1.0)) * 0.5) / abs(J)), U, 1.0)) * cos(Float64(K * -0.5))) * t_1);
	else
		tmp = Float64(t_3 * Float64(abs(Float64(t_5 * U)) / abs(J)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(t$95$1 * t$95$2), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(0.5 / N[Abs[t$95$0], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, (-Infinity)], N[(N[(N[(t$95$5 * N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0), $MachinePrecision] * N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$4, 4e+293], N[(N[(N[Sqrt[N[(N[(N[(N[(N[(U / N[Abs[J], $MachinePrecision]), $MachinePrecision] / N[(N[Cos[K], $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * 0.5), $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision] * U + 1.0), $MachinePrecision]], $MachinePrecision] * N[Cos[N[(K * -0.5), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], N[(t$95$3 * N[(N[Abs[N[(t$95$5 * U), $MachinePrecision]], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{\color{blue}{J}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      8. lower-*.f64 14.1%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   4. Applied rewrites 14.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J}} \]

   6. Applied rewrites 20.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\left|U\right| \cdot \sqrt{\frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)} \cdot 0.25}}{J} \]

   8. Applied rewrites 21.0%

      \[\leadsto \color{blue}{\left(\left(\frac{0.5}{\left|\cos \left(0.5 \cdot K\right)\right|} \cdot \frac{\left|U\right|}{J}\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right)} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 3.9999999999999997e293

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   3. Applied rewrites 73.7%

      \[\leadsto \color{blue}{\left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{0.5 + 0.5 \cdot \cos K} - -1}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\color{blue}{\frac{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}} - -1} \]

      2. lift-/.f64 N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{\color{blue}{\frac{\frac{U}{J} \cdot \frac{U}{J}}{4}}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K} - -1} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{\frac{\color{blue}{\frac{U}{J} \cdot \frac{U}{J}}}{4}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K} - -1} \]

      4. associate-/l* N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{\color{blue}{\frac{U}{J} \cdot \frac{\frac{U}{J}}{4}}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K} - -1} \]

      5. associate-/l* N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\color{blue}{\frac{U}{J} \cdot \frac{\frac{\frac{U}{J}}{4}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K}} - -1} \]

      6. lift-/.f64 N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\color{blue}{\frac{U}{J}} \cdot \frac{\frac{\frac{U}{J}}{4}}{\frac{1}{2} + \frac{1}{2} \cdot \cos K} - -1} \]

      7. frac-times N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\color{blue}{\frac{U \cdot \frac{\frac{U}{J}}{4}}{J \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos K\right)}} - -1} \]

      8. lower-/.f64 N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\color{blue}{\frac{U \cdot \frac{\frac{U}{J}}{4}}{J \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos K\right)}} - -1} \]

      9. lower-*.f64 N/A

         \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{\color{blue}{U \cdot \frac{\frac{U}{J}}{4}}}{J \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos K\right)} - -1} \]

      10. mult-flip N/A

          \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U \cdot \color{blue}{\left(\frac{U}{J} \cdot \frac{1}{4}\right)}}{J \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos K\right)} - -1} \]

      11. metadata-eval N/A

          \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U \cdot \left(\frac{U}{J} \cdot \color{blue}{\frac{1}{4}}\right)}{J \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos K\right)} - -1} \]

      12. lower-*.f64 N/A

          \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U \cdot \color{blue}{\left(\frac{U}{J} \cdot \frac{1}{4}\right)}}{J \cdot \left(\frac{1}{2} + \frac{1}{2} \cdot \cos K\right)} - -1} \]

      13. lower-*.f64 70.8%

          \[\leadsto \left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U \cdot \left(\frac{U}{J} \cdot 0.25\right)}{\color{blue}{J \cdot \left(0.5 + 0.5 \cdot \cos K\right)}} - -1} \]

      14. lift-+.f64 N/A

          \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U \cdot \left(\frac{U}{J} \cdot \frac{1}{4}\right)}{J \cdot \color{blue}{\left(\frac{1}{2} + \frac{1}{2} \cdot \cos K\right)}} - -1} \]

      15. +-commutative N/A

          \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U \cdot \left(\frac{U}{J} \cdot \frac{1}{4}\right)}{J \cdot \color{blue}{\left(\frac{1}{2} \cdot \cos K + \frac{1}{2}\right)}} - -1} \]

      16. lift-*.f64 N/A

          \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U \cdot \left(\frac{U}{J} \cdot \frac{1}{4}\right)}{J \cdot \left(\color{blue}{\frac{1}{2} \cdot \cos K} + \frac{1}{2}\right)} - -1} \]

      17. *-commutative N/A

          \[\leadsto \left(\left(\cos \left(\frac{-1}{2} \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U \cdot \left(\frac{U}{J} \cdot \frac{1}{4}\right)}{J \cdot \left(\color{blue}{\cos K \cdot \frac{1}{2}} + \frac{1}{2}\right)} - -1} \]

      18. lower-fma.f64 70.8%

          \[\leadsto \left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\frac{U \cdot \left(\frac{U}{J} \cdot 0.25\right)}{J \cdot \color{blue}{\mathsf{fma}\left(\cos K, 0.5, 0.5\right)}} - -1} \]

   5. Applied rewrites 70.8%

      \[\leadsto \left(\left(\cos \left(-0.5 \cdot K\right) \cdot -2\right) \cdot J\right) \cdot \sqrt{\color{blue}{\frac{U \cdot \left(\frac{U}{J} \cdot 0.25\right)}{J \cdot \mathsf{fma}\left(\cos K, 0.5, 0.5\right)}} - -1} \]

   7. Applied rewrites 70.5%

      \[\leadsto \color{blue}{\left(\sqrt{\mathsf{fma}\left(\frac{\frac{\frac{U}{J}}{\cos K - -1} \cdot 0.5}{J}, U, 1\right)} \cdot \cos \left(K \cdot -0.5\right)\right) \cdot \left(-2 \cdot J\right)} \]

   if 3.9999999999999997e293 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{\color{blue}{J}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      8. lower-*.f64 14.1%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   4. Applied rewrites 14.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J}} \]

   6. Applied rewrites 20.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\left|U\right| \cdot \sqrt{\frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)} \cdot 0.25}}{J} \]

   8. Applied rewrites 20.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\left|\frac{0.5}{\left|\cos \left(0.5 \cdot K\right)\right|} \cdot U\right|}{\color{blue}{J}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 74.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{U}{\left|J\right| + \left|J\right|}\\ t_1 := \cos \left(0.5 \cdot K\right)\\ t_2 := -2 \cdot \left|J\right|\\ t_3 := \cos \left(\frac{K}{2}\right)\\ t_4 := t\_2 \cdot t\_3\\ t_5 := t\_4 \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_3}\right)}^{2}}\\ t_6 := \frac{0.5}{\left|t\_1\right|}\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_5 \leq -\infty:\\ \;\;\;\;\left(\left(t\_6 \cdot \frac{\left|U\right|}{\left|J\right|}\right) \cdot t\_1\right) \cdot \left(\left|J\right| \cdot -2\right)\\ \mathbf{elif}\;t\_5 \leq 5 \cdot 10^{+125}:\\ \;\;\;\;\left(\cos \left(K \cdot -0.5\right) \cdot \sqrt{\mathsf{fma}\left(t\_0, t\_0, 1\right)}\right) \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_4 \cdot \frac{\left|t\_6 \cdot U\right|}{\left|J\right|}\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (/ U (+ (fabs J) (fabs J))))
        (t_1 (cos (* 0.5 K)))
        (t_2 (* -2.0 (fabs J)))
        (t_3 (cos (/ K 2.0)))
        (t_4 (* t_2 t_3))
        (t_5 (* t_4 (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_3)) 2.0)))))
        (t_6 (/ 0.5 (fabs t_1))))
   (*
    (copysign 1.0 J)
    (if (<= t_5 (- INFINITY))
      (* (* (* t_6 (/ (fabs U) (fabs J))) t_1) (* (fabs J) -2.0))
      (if (<= t_5 5e+125)
        (* (* (cos (* K -0.5)) (sqrt (fma t_0 t_0 1.0))) t_2)
        (* t_4 (/ (fabs (* t_6 U)) (fabs J))))))))

C

double code(double J, double K, double U) {
	double t_0 = U / (fabs(J) + fabs(J));
	double t_1 = cos((0.5 * K));
	double t_2 = -2.0 * fabs(J);
	double t_3 = cos((K / 2.0));
	double t_4 = t_2 * t_3;
	double t_5 = t_4 * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_3)), 2.0)));
	double t_6 = 0.5 / fabs(t_1);
	double tmp;
	if (t_5 <= -((double) INFINITY)) {
		tmp = ((t_6 * (fabs(U) / fabs(J))) * t_1) * (fabs(J) * -2.0);
	} else if (t_5 <= 5e+125) {
		tmp = (cos((K * -0.5)) * sqrt(fma(t_0, t_0, 1.0))) * t_2;
	} else {
		tmp = t_4 * (fabs((t_6 * U)) / fabs(J));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = Float64(U / Float64(abs(J) + abs(J)))
	t_1 = cos(Float64(0.5 * K))
	t_2 = Float64(-2.0 * abs(J))
	t_3 = cos(Float64(K / 2.0))
	t_4 = Float64(t_2 * t_3)
	t_5 = Float64(t_4 * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_3)) ^ 2.0))))
	t_6 = Float64(0.5 / abs(t_1))
	tmp = 0.0
	if (t_5 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(t_6 * Float64(abs(U) / abs(J))) * t_1) * Float64(abs(J) * -2.0));
	elseif (t_5 <= 5e+125)
		tmp = Float64(Float64(cos(Float64(K * -0.5)) * sqrt(fma(t_0, t_0, 1.0))) * t_2);
	else
		tmp = Float64(t_4 * Float64(abs(Float64(t_6 * U)) / abs(J)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(U / N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$4 = N[(t$95$2 * t$95$3), $MachinePrecision]}, Block[{t$95$5 = N[(t$95$4 * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$6 = N[(0.5 / N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$5, (-Infinity)], N[(N[(N[(t$95$6 * N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$5, 5e+125], N[(N[(N[Cos[N[(K * -0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision], N[(t$95$4 * N[(N[Abs[N[(t$95$6 * U), $MachinePrecision]], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{\color{blue}{J}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      8. lower-*.f64 14.1%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   4. Applied rewrites 14.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J}} \]

   6. Applied rewrites 20.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\left|U\right| \cdot \sqrt{\frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)} \cdot 0.25}}{J} \]

   8. Applied rewrites 21.0%

      \[\leadsto \color{blue}{\left(\left(\frac{0.5}{\left|\cos \left(0.5 \cdot K\right)\right|} \cdot \frac{\left|U\right|}{J}\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right)} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.99999999999999962e125

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} + 1} \]

      5. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}} + 1} \]

      6. cosh-asinh-rev N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      7. lower-cosh.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      8. lower-asinh.f64 85.0%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \color{blue}{\sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(2 \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      10. count-2-rev N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(J + J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      11. lower-+.f64 85.0%

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(J + J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      12. lift-cos.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}}\right) \]

      13. cos-neg-rev N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}}\right) \]

      14. lower-cos.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right)}\right) \]

      16. distribute-neg-frac2 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{\color{blue}{-2}}\right)}\right) \]

      18. mult-flip-rev N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{-2}\right)}}\right) \]

      19. *-commutative N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{1}{-2} \cdot K\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{1}{-2} \cdot K\right)}}\right) \]

      21. metadata-eval 85.0%

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\color{blue}{-0.5} \cdot K\right)}\right) \]

   3. Applied rewrites 85.0%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right)} \]

   4. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{U}{J}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \color{blue}{\frac{U}{J}}\right) \]

      2. lower-/.f64 71.0%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(0.5 \cdot \frac{U}{\color{blue}{J}}\right) \]

   6. Applied rewrites 71.0%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \color{blue}{\left(0.5 \cdot \frac{U}{J}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \frac{U}{J}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)} \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \frac{U}{J}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \frac{U}{J}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{K}{2}\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \frac{U}{J}\right)\right) \cdot \left(-2 \cdot J\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{K}{2}\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \frac{U}{J}\right)\right) \cdot \left(-2 \cdot J\right)} \]

   8. Applied rewrites 64.4%

      \[\leadsto \color{blue}{\left(\cos \left(K \cdot -0.5\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J + J}, \frac{U}{J + J}, 1\right)}\right) \cdot \left(-2 \cdot J\right)} \]

   if 4.99999999999999962e125 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{\color{blue}{J}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      8. lower-*.f64 14.1%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   4. Applied rewrites 14.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J}} \]

   6. Applied rewrites 20.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\left|U\right| \cdot \sqrt{\frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)} \cdot 0.25}}{J} \]

   8. Applied rewrites 20.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\left|\frac{0.5}{\left|\cos \left(0.5 \cdot K\right)\right|} \cdot U\right|}{\color{blue}{J}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{U}{\left|J\right| + \left|J\right|}\\ t_1 := \cos \left(0.5 \cdot K\right)\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := -2 \cdot \left|J\right|\\ t_4 := \left(t\_3 \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ t_5 := \left(\left(\frac{0.5}{\left|t\_1\right|} \cdot \frac{\left|U\right|}{\left|J\right|}\right) \cdot t\_1\right) \cdot \left(\left|J\right| \cdot -2\right)\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq 5 \cdot 10^{+125}:\\ \;\;\;\;\left(\cos \left(K \cdot -0.5\right) \cdot \sqrt{\mathsf{fma}\left(t\_0, t\_0, 1\right)}\right) \cdot t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (/ U (+ (fabs J) (fabs J))))
        (t_1 (cos (* 0.5 K)))
        (t_2 (cos (/ K 2.0)))
        (t_3 (* -2.0 (fabs J)))
        (t_4
         (*
          (* t_3 t_2)
          (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_2)) 2.0)))))
        (t_5
         (*
          (* (* (/ 0.5 (fabs t_1)) (/ (fabs U) (fabs J))) t_1)
          (* (fabs J) -2.0))))
   (*
    (copysign 1.0 J)
    (if (<= t_4 (- INFINITY))
      t_5
      (if (<= t_4 5e+125)
        (* (* (cos (* K -0.5)) (sqrt (fma t_0 t_0 1.0))) t_3)
        t_5)))))

C

double code(double J, double K, double U) {
	double t_0 = U / (fabs(J) + fabs(J));
	double t_1 = cos((0.5 * K));
	double t_2 = cos((K / 2.0));
	double t_3 = -2.0 * fabs(J);
	double t_4 = (t_3 * t_2) * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_2)), 2.0)));
	double t_5 = (((0.5 / fabs(t_1)) * (fabs(U) / fabs(J))) * t_1) * (fabs(J) * -2.0);
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_5;
	} else if (t_4 <= 5e+125) {
		tmp = (cos((K * -0.5)) * sqrt(fma(t_0, t_0, 1.0))) * t_3;
	} else {
		tmp = t_5;
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = Float64(U / Float64(abs(J) + abs(J)))
	t_1 = cos(Float64(0.5 * K))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(-2.0 * abs(J))
	t_4 = Float64(Float64(t_3 * t_2) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_2)) ^ 2.0))))
	t_5 = Float64(Float64(Float64(Float64(0.5 / abs(t_1)) * Float64(abs(U) / abs(J))) * t_1) * Float64(abs(J) * -2.0))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_4 <= 5e+125)
		tmp = Float64(Float64(cos(Float64(K * -0.5)) * sqrt(fma(t_0, t_0, 1.0))) * t_3);
	else
		tmp = t_5;
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(U / N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(N[(t$95$3 * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(N[(0.5 / N[Abs[t$95$1], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision] * N[(N[Abs[J], $MachinePrecision] * -2.0), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$4, (-Infinity)], t$95$5, If[LessEqual[t$95$4, 5e+125], N[(N[(N[Cos[N[(K * -0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$3), $MachinePrecision], t$95$5]]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 4.99999999999999962e125 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{\color{blue}{J}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      8. lower-*.f64 14.1%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   4. Applied rewrites 14.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J}} \]

   6. Applied rewrites 20.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\left|U\right| \cdot \sqrt{\frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)} \cdot 0.25}}{J} \]

   8. Applied rewrites 21.0%

      \[\leadsto \color{blue}{\left(\left(\frac{0.5}{\left|\cos \left(0.5 \cdot K\right)\right|} \cdot \frac{\left|U\right|}{J}\right) \cdot \cos \left(0.5 \cdot K\right)\right) \cdot \left(J \cdot -2\right)} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.99999999999999962e125

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} + 1} \]

      5. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}} + 1} \]

      6. cosh-asinh-rev N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      7. lower-cosh.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      8. lower-asinh.f64 85.0%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \color{blue}{\sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(2 \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      10. count-2-rev N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(J + J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      11. lower-+.f64 85.0%

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(J + J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      12. lift-cos.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}}\right) \]

      13. cos-neg-rev N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}}\right) \]

      14. lower-cos.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right)}\right) \]

      16. distribute-neg-frac2 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{\color{blue}{-2}}\right)}\right) \]

      18. mult-flip-rev N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{-2}\right)}}\right) \]

      19. *-commutative N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{1}{-2} \cdot K\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{1}{-2} \cdot K\right)}}\right) \]

      21. metadata-eval 85.0%

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\color{blue}{-0.5} \cdot K\right)}\right) \]

   3. Applied rewrites 85.0%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right)} \]

   4. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{U}{J}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \color{blue}{\frac{U}{J}}\right) \]

      2. lower-/.f64 71.0%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(0.5 \cdot \frac{U}{\color{blue}{J}}\right) \]

   6. Applied rewrites 71.0%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \color{blue}{\left(0.5 \cdot \frac{U}{J}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \frac{U}{J}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)} \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \frac{U}{J}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \frac{U}{J}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{K}{2}\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \frac{U}{J}\right)\right) \cdot \left(-2 \cdot J\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{K}{2}\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \frac{U}{J}\right)\right) \cdot \left(-2 \cdot J\right)} \]

   8. Applied rewrites 64.4%

      \[\leadsto \color{blue}{\left(\cos \left(K \cdot -0.5\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J + J}, \frac{U}{J + J}, 1\right)}\right) \cdot \left(-2 \cdot J\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 74.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_0 := \frac{U}{\left|J\right| + \left|J\right|}\\ t_1 := -2 \cdot \left|J\right|\\ t_2 := \cos \left(\frac{K}{2}\right)\\ t_3 := \left(t\_1 \cdot t\_2\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_2}\right)}^{2}}\\ t_4 := \cos \left(0.5 \cdot K\right)\\ t_5 := \left(\left(-2 \cdot t\_4\right) \cdot \left|J\right|\right) \cdot \left(\frac{0.5}{\left|t\_4\right|} \cdot \frac{\left|U\right|}{\left|J\right|}\right)\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{+125}:\\ \;\;\;\;\left(\cos \left(K \cdot -0.5\right) \cdot \sqrt{\mathsf{fma}\left(t\_0, t\_0, 1\right)}\right) \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (/ U (+ (fabs J) (fabs J))))
        (t_1 (* -2.0 (fabs J)))
        (t_2 (cos (/ K 2.0)))
        (t_3
         (*
          (* t_1 t_2)
          (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_2)) 2.0)))))
        (t_4 (cos (* 0.5 K)))
        (t_5
         (*
          (* (* -2.0 t_4) (fabs J))
          (* (/ 0.5 (fabs t_4)) (/ (fabs U) (fabs J))))))
   (*
    (copysign 1.0 J)
    (if (<= t_3 (- INFINITY))
      t_5
      (if (<= t_3 5e+125)
        (* (* (cos (* K -0.5)) (sqrt (fma t_0 t_0 1.0))) t_1)
        t_5)))))

C

double code(double J, double K, double U) {
	double t_0 = U / (fabs(J) + fabs(J));
	double t_1 = -2.0 * fabs(J);
	double t_2 = cos((K / 2.0));
	double t_3 = (t_1 * t_2) * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_2)), 2.0)));
	double t_4 = cos((0.5 * K));
	double t_5 = ((-2.0 * t_4) * fabs(J)) * ((0.5 / fabs(t_4)) * (fabs(U) / fabs(J)));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_5;
	} else if (t_3 <= 5e+125) {
		tmp = (cos((K * -0.5)) * sqrt(fma(t_0, t_0, 1.0))) * t_1;
	} else {
		tmp = t_5;
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = Float64(U / Float64(abs(J) + abs(J)))
	t_1 = Float64(-2.0 * abs(J))
	t_2 = cos(Float64(K / 2.0))
	t_3 = Float64(Float64(t_1 * t_2) * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_2)) ^ 2.0))))
	t_4 = cos(Float64(0.5 * K))
	t_5 = Float64(Float64(Float64(-2.0 * t_4) * abs(J)) * Float64(Float64(0.5 / abs(t_4)) * Float64(abs(U) / abs(J))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_3 <= 5e+125)
		tmp = Float64(Float64(cos(Float64(K * -0.5)) * sqrt(fma(t_0, t_0, 1.0))) * t_1);
	else
		tmp = t_5;
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(U / N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$3 = N[(N[(t$95$1 * t$95$2), $MachinePrecision] * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[Cos[N[(0.5 * K), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$5 = N[(N[(N[(-2.0 * t$95$4), $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * N[(N[(0.5 / N[Abs[t$95$4], $MachinePrecision]), $MachinePrecision] * N[(N[Abs[U], $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[t$95$3, (-Infinity)], t$95$5, If[LessEqual[t$95$3, 5e+125], N[(N[(N[Cos[N[(K * -0.5), $MachinePrecision]], $MachinePrecision] * N[Sqrt[N[(t$95$0 * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision], t$95$5]]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0 or 4.99999999999999962e125 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{\color{blue}{J}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      8. lower-*.f64 14.1%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   4. Applied rewrites 14.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J}} \]

   6. Applied rewrites 20.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\left|U\right| \cdot \sqrt{\frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)} \cdot 0.25}}{J} \]

   8. Applied rewrites 21.0%

      \[\leadsto \color{blue}{\left(\left(-2 \cdot \cos \left(0.5 \cdot K\right)\right) \cdot J\right) \cdot \left(\frac{0.5}{\left|\cos \left(0.5 \cdot K\right)\right|} \cdot \frac{\left|U\right|}{J}\right)} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < 4.99999999999999962e125

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} + 1} \]

      5. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}} + 1} \]

      6. cosh-asinh-rev N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      7. lower-cosh.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      8. lower-asinh.f64 85.0%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \color{blue}{\sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(2 \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      10. count-2-rev N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(J + J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      11. lower-+.f64 85.0%

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(J + J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      12. lift-cos.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}}\right) \]

      13. cos-neg-rev N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}}\right) \]

      14. lower-cos.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right)}\right) \]

      16. distribute-neg-frac2 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{\color{blue}{-2}}\right)}\right) \]

      18. mult-flip-rev N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{-2}\right)}}\right) \]

      19. *-commutative N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{1}{-2} \cdot K\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{1}{-2} \cdot K\right)}}\right) \]

      21. metadata-eval 85.0%

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\color{blue}{-0.5} \cdot K\right)}\right) \]

   3. Applied rewrites 85.0%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right)} \]

   4. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{U}{J}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \color{blue}{\frac{U}{J}}\right) \]

      2. lower-/.f64 71.0%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(0.5 \cdot \frac{U}{\color{blue}{J}}\right) \]

   6. Applied rewrites 71.0%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \color{blue}{\left(0.5 \cdot \frac{U}{J}\right)} \]
   7. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \frac{U}{J}\right)} \]

      2. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right)} \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \frac{U}{J}\right) \]

      3. associate-*l* N/A

         \[\leadsto \color{blue}{\left(-2 \cdot J\right) \cdot \left(\cos \left(\frac{K}{2}\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \frac{U}{J}\right)\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{K}{2}\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \frac{U}{J}\right)\right) \cdot \left(-2 \cdot J\right)} \]

      5. lower-*.f64 N/A

         \[\leadsto \color{blue}{\left(\cos \left(\frac{K}{2}\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \frac{U}{J}\right)\right) \cdot \left(-2 \cdot J\right)} \]

   8. Applied rewrites 64.4%

      \[\leadsto \color{blue}{\left(\cos \left(K \cdot -0.5\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J + J}, \frac{U}{J + J}, 1\right)}\right) \cdot \left(-2 \cdot J\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 10: 71.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} t_0 := \frac{U}{\left|J\right| + \left|J\right|}\\ t_1 := \cos \left(\frac{K}{2}\right)\\ t_2 := \left(-2 \cdot \left|J\right|\right) \cdot t\_1\\ \mathsf{copysign}\left(1, J\right) \cdot \begin{array}{l} \mathbf{if}\;t\_2 \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot \left|J\right|\right) \cdot t\_1}\right)}^{2}} \leq -\infty:\\ \;\;\;\;t\_2 \cdot \frac{\left|U\right| \cdot 0.5}{\left|J\right|}\\ \mathbf{else}:\\ \;\;\;\;\left(\left(\cos \left(K \cdot -0.5\right) \cdot \left|J\right|\right) \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(t\_0, t\_0, 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (/ U (+ (fabs J) (fabs J))))
        (t_1 (cos (/ K 2.0)))
        (t_2 (* (* -2.0 (fabs J)) t_1)))
   (*
    (copysign 1.0 J)
    (if (<=
         (* t_2 (sqrt (+ 1.0 (pow (/ U (* (* 2.0 (fabs J)) t_1)) 2.0))))
         (- INFINITY))
      (* t_2 (/ (* (fabs U) 0.5) (fabs J)))
      (* (* (* (cos (* K -0.5)) (fabs J)) -2.0) (sqrt (fma t_0 t_0 1.0)))))))

C

double code(double J, double K, double U) {
	double t_0 = U / (fabs(J) + fabs(J));
	double t_1 = cos((K / 2.0));
	double t_2 = (-2.0 * fabs(J)) * t_1;
	double tmp;
	if ((t_2 * sqrt((1.0 + pow((U / ((2.0 * fabs(J)) * t_1)), 2.0)))) <= -((double) INFINITY)) {
		tmp = t_2 * ((fabs(U) * 0.5) / fabs(J));
	} else {
		tmp = ((cos((K * -0.5)) * fabs(J)) * -2.0) * sqrt(fma(t_0, t_0, 1.0));
	}
	return copysign(1.0, J) * tmp;
}

Julia

function code(J, K, U)
	t_0 = Float64(U / Float64(abs(J) + abs(J)))
	t_1 = cos(Float64(K / 2.0))
	t_2 = Float64(Float64(-2.0 * abs(J)) * t_1)
	tmp = 0.0
	if (Float64(t_2 * sqrt(Float64(1.0 + (Float64(U / Float64(Float64(2.0 * abs(J)) * t_1)) ^ 2.0)))) <= Float64(-Inf))
		tmp = Float64(t_2 * Float64(Float64(abs(U) * 0.5) / abs(J)));
	else
		tmp = Float64(Float64(Float64(cos(Float64(K * -0.5)) * abs(J)) * -2.0) * sqrt(fma(t_0, t_0, 1.0)));
	end
	return Float64(copysign(1.0, J) * tmp)
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(U / N[(N[Abs[J], $MachinePrecision] + N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[Cos[N[(K / 2.0), $MachinePrecision]], $MachinePrecision]}, Block[{t$95$2 = N[(N[(-2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]}, N[(N[With[{TMP1 = Abs[1.0], TMP2 = Sign[J]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision] * If[LessEqual[N[(t$95$2 * N[Sqrt[N[(1.0 + N[Power[N[(U / N[(N[(2.0 * N[Abs[J], $MachinePrecision]), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], (-Infinity)], N[(t$95$2 * N[(N[(N[Abs[U], $MachinePrecision] * 0.5), $MachinePrecision] / N[Abs[J], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(N[Cos[N[(K * -0.5), $MachinePrecision]], $MachinePrecision] * N[Abs[J], $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] * N[Sqrt[N[(t$95$0 * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64))))) < -inf.0

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{\color{blue}{J}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      8. lower-*.f64 14.1%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   4. Applied rewrites 14.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J}} \]

   6. Applied rewrites 20.9%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\left|U\right| \cdot \sqrt{\frac{1}{0.5 + 0.5 \cdot \cos \left(2 \cdot \left(-0.5 \cdot K\right)\right)} \cdot 0.25}}{J} \]

   7. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\left|U\right| \cdot \frac{1}{2}}{J} \]
   8. Step-by-step derivation

   9. Applied rewrites 13.3%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\left|U\right| \cdot 0.5}{J} \]

   if -inf.0 < (*.f64 (*.f64 (*.f64 #s(literal -2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64)))) (sqrt.f64 (+.f64 #s(literal 1 binary64) (pow.f64 (/.f64 U (*.f64 (*.f64 #s(literal 2 binary64) J) (cos.f64 (/.f64 K #s(literal 2 binary64))))) #s(literal 2 binary64)))))

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} + 1} \]

      5. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}} + 1} \]

      6. cosh-asinh-rev N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      7. lower-cosh.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      8. lower-asinh.f64 85.0%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \color{blue}{\sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(2 \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      10. count-2-rev N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(J + J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      11. lower-+.f64 85.0%

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(J + J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      12. lift-cos.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}}\right) \]

      13. cos-neg-rev N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}}\right) \]

      14. lower-cos.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right)}\right) \]

      16. distribute-neg-frac2 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{\color{blue}{-2}}\right)}\right) \]

      18. mult-flip-rev N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{-2}\right)}}\right) \]

      19. *-commutative N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{1}{-2} \cdot K\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{1}{-2} \cdot K\right)}}\right) \]

      21. metadata-eval 85.0%

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\color{blue}{-0.5} \cdot K\right)}\right) \]

   3. Applied rewrites 85.0%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right)} \]

   4. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{U}{J}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \color{blue}{\frac{U}{J}}\right) \]

      2. lower-/.f64 71.0%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(0.5 \cdot \frac{U}{\color{blue}{J}}\right) \]

   6. Applied rewrites 71.0%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \color{blue}{\left(0.5 \cdot \frac{U}{J}\right)} \]

   8. Applied rewrites 64.4%

      \[\leadsto \color{blue}{\left(\left(\cos \left(K \cdot -0.5\right) \cdot J\right) \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J + J}, \frac{U}{J + J}, 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 11: 71.0% accurate, 1.6× speedup

Math

\[\left(\left(\cos \left(K \cdot -0.5\right) \cdot J\right) \cdot -2\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{J + J}\right) \]

FPCore

(FPCore (J K U)
 :precision binary64
 (* (* (* (cos (* K -0.5)) J) -2.0) (cosh (asinh (/ U (+ J J))))))

C

double code(double J, double K, double U) {
	return ((cos((K * -0.5)) * J) * -2.0) * cosh(asinh((U / (J + J))));
}

Python

def code(J, K, U):
	return ((math.cos((K * -0.5)) * J) * -2.0) * math.cosh(math.asinh((U / (J + J))))

Julia

function code(J, K, U)
	return Float64(Float64(Float64(cos(Float64(K * -0.5)) * J) * -2.0) * cosh(asinh(Float64(U / Float64(J + J)))))
end

MATLAB

function tmp = code(J, K, U)
	tmp = ((cos((K * -0.5)) * J) * -2.0) * cosh(asinh((U / (J + J))));
end

Wolfram

code[J_, K_, U_] := N[(N[(N[(N[Cos[N[(K * -0.5), $MachinePrecision]], $MachinePrecision] * J), $MachinePrecision] * -2.0), $MachinePrecision] * N[Cosh[N[ArcSinh[N[(U / N[(J + J), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.8%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
2. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}} \]

   2. lift-+.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}} \]

   3. +-commutative N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1}} \]

   4. lift-pow.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} + 1} \]

   5. unpow2 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}} + 1} \]

   6. cosh-asinh-rev N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

   7. lower-cosh.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

   8. lower-asinh.f64 85.0%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \color{blue}{\sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

   9. lift-*.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(2 \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

   10. count-2-rev N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(J + J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

   11. lower-+.f64 85.0%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(J + J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

   12. lift-cos.f64 N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}}\right) \]

   13. cos-neg-rev N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}}\right) \]

   14. lower-cos.f64 N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}}\right) \]

   15. lift-/.f64 N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right)}\right) \]

   16. distribute-neg-frac2 N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}}\right) \]

   17. metadata-eval N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{\color{blue}{-2}}\right)}\right) \]

   18. mult-flip-rev N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{-2}\right)}}\right) \]

   19. *-commutative N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{1}{-2} \cdot K\right)}}\right) \]

   20. lower-*.f64 N/A

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{1}{-2} \cdot K\right)}}\right) \]

   21. metadata-eval 85.0%

       \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\color{blue}{-0.5} \cdot K\right)}\right) \]

3. Applied rewrites 85.0%

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right)} \]

4. Taylor expanded in K around 0

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{U}{J}\right)} \]
5. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \color{blue}{\frac{U}{J}}\right) \]

   2. lower-/.f64 71.0%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(0.5 \cdot \frac{U}{\color{blue}{J}}\right) \]

6. Applied rewrites 71.0%

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \color{blue}{\left(0.5 \cdot \frac{U}{J}\right)} \]

8. Applied rewrites 64.4%

   \[\leadsto \color{blue}{\left(\left(\cos \left(K \cdot -0.5\right) \cdot J\right) \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J + J}, \frac{U}{J + J}, 1\right)}} \]
9. Step-by-step derivation

   1. lift-sqrt.f64 N/A

      \[\leadsto \left(\left(\cos \left(K \cdot \frac{-1}{2}\right) \cdot J\right) \cdot -2\right) \cdot \color{blue}{\sqrt{\mathsf{fma}\left(\frac{U}{J + J}, \frac{U}{J + J}, 1\right)}} \]

   2. lift-fma.f64 N/A

      \[\leadsto \left(\left(\cos \left(K \cdot \frac{-1}{2}\right) \cdot J\right) \cdot -2\right) \cdot \sqrt{\color{blue}{\frac{U}{J + J} \cdot \frac{U}{J + J} + 1}} \]

   3. cosh-asinh-rev N/A

      \[\leadsto \left(\left(\cos \left(K \cdot \frac{-1}{2}\right) \cdot J\right) \cdot -2\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{J + J}\right)} \]

   4. lower-cosh.f64 N/A

      \[\leadsto \left(\left(\cos \left(K \cdot \frac{-1}{2}\right) \cdot J\right) \cdot -2\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{J + J}\right)} \]

   5. lower-asinh.f64 71.0%

      \[\leadsto \left(\left(\cos \left(K \cdot -0.5\right) \cdot J\right) \cdot -2\right) \cdot \cosh \color{blue}{\sinh^{-1} \left(\frac{U}{J + J}\right)} \]

10. Applied rewrites 71.0%

    \[\leadsto \left(\left(\cos \left(K \cdot -0.5\right) \cdot J\right) \cdot -2\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{J + J}\right)} \]
11. Add Preprocessing

Alternative 12: 62.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_0 := \frac{U}{J + J}\\ \mathbf{if}\;\left|K\right| \leq 420:\\ \;\;\;\;\left(\left(J + -0.125 \cdot \left(J \cdot {\left(\left|K\right|\right)}^{2}\right)\right) \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(t\_0, t\_0, 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\cos \left(-0.5 \cdot \left|K\right|\right) \cdot \left(-2 \cdot J\right)\\ \end{array} \]

FPCore

(FPCore (J K U)
 :precision binary64
 (let* ((t_0 (/ U (+ J J))))
   (if (<= (fabs K) 420.0)
     (*
      (* (+ J (* -0.125 (* J (pow (fabs K) 2.0)))) -2.0)
      (sqrt (fma t_0 t_0 1.0)))
     (* (cos (* -0.5 (fabs K))) (* -2.0 J)))))

C

double code(double J, double K, double U) {
	double t_0 = U / (J + J);
	double tmp;
	if (fabs(K) <= 420.0) {
		tmp = ((J + (-0.125 * (J * pow(fabs(K), 2.0)))) * -2.0) * sqrt(fma(t_0, t_0, 1.0));
	} else {
		tmp = cos((-0.5 * fabs(K))) * (-2.0 * J);
	}
	return tmp;
}

Julia

function code(J, K, U)
	t_0 = Float64(U / Float64(J + J))
	tmp = 0.0
	if (abs(K) <= 420.0)
		tmp = Float64(Float64(Float64(J + Float64(-0.125 * Float64(J * (abs(K) ^ 2.0)))) * -2.0) * sqrt(fma(t_0, t_0, 1.0)));
	else
		tmp = Float64(cos(Float64(-0.5 * abs(K))) * Float64(-2.0 * J));
	end
	return tmp
end

Wolfram

code[J_, K_, U_] := Block[{t$95$0 = N[(U / N[(J + J), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[K], $MachinePrecision], 420.0], N[(N[(N[(J + N[(-0.125 * N[(J * N[Power[N[Abs[K], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * -2.0), $MachinePrecision] * N[Sqrt[N[(t$95$0 * t$95$0 + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision], N[(N[Cos[N[(-0.5 * N[Abs[K], $MachinePrecision]), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if K < 420

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]
   2. Step-by-step derivation

      1. lift-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}} \]

      2. lift-+.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}}} \]

      3. +-commutative N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2} + 1}} \]

      4. lift-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{{\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} + 1} \]

      5. unpow2 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{\color{blue}{\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)} \cdot \frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}} + 1} \]

      6. cosh-asinh-rev N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      7. lower-cosh.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      8. lower-asinh.f64 85.0%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \color{blue}{\sinh^{-1} \left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)} \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(2 \cdot J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      10. count-2-rev N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(J + J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      11. lower-+.f64 85.0%

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\color{blue}{\left(J + J\right)} \cdot \cos \left(\frac{K}{2}\right)}\right) \]

      12. lift-cos.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\frac{K}{2}\right)}}\right) \]

      13. cos-neg-rev N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}}\right) \]

      14. lower-cos.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \color{blue}{\cos \left(\mathsf{neg}\left(\frac{K}{2}\right)\right)}}\right) \]

      15. lift-/.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\mathsf{neg}\left(\color{blue}{\frac{K}{2}}\right)\right)}\right) \]

      16. distribute-neg-frac2 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{K}{\mathsf{neg}\left(2\right)}\right)}}\right) \]

      17. metadata-eval N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\frac{K}{\color{blue}{-2}}\right)}\right) \]

      18. mult-flip-rev N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(K \cdot \frac{1}{-2}\right)}}\right) \]

      19. *-commutative N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{1}{-2} \cdot K\right)}}\right) \]

      20. lower-*.f64 N/A

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \color{blue}{\left(\frac{1}{-2} \cdot K\right)}}\right) \]

      21. metadata-eval 85.0%

          \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(\color{blue}{-0.5} \cdot K\right)}\right) \]

   3. Applied rewrites 85.0%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\cosh \sinh^{-1} \left(\frac{U}{\left(J + J\right) \cdot \cos \left(-0.5 \cdot K\right)}\right)} \]

   4. Taylor expanded in K around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \color{blue}{\left(\frac{1}{2} \cdot \frac{U}{J}\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(\frac{1}{2} \cdot \color{blue}{\frac{U}{J}}\right) \]

      2. lower-/.f64 71.0%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \left(0.5 \cdot \frac{U}{\color{blue}{J}}\right) \]

   6. Applied rewrites 71.0%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \cosh \sinh^{-1} \color{blue}{\left(0.5 \cdot \frac{U}{J}\right)} \]

   8. Applied rewrites 64.4%

      \[\leadsto \color{blue}{\left(\left(\cos \left(K \cdot -0.5\right) \cdot J\right) \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J + J}, \frac{U}{J + J}, 1\right)}} \]

   9. Taylor expanded in K around 0

      \[\leadsto \left(\color{blue}{\left(J + \frac{-1}{8} \cdot \left(J \cdot {K}^{2}\right)\right)} \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J + J}, \frac{U}{J + J}, 1\right)} \]
   10. Step-by-step derivation

       1. lower-+.f64 N/A

          \[\leadsto \left(\left(J + \color{blue}{\frac{-1}{8} \cdot \left(J \cdot {K}^{2}\right)}\right) \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J + J}, \frac{U}{J + J}, 1\right)} \]

       2. lower-*.f64 N/A

          \[\leadsto \left(\left(J + \frac{-1}{8} \cdot \color{blue}{\left(J \cdot {K}^{2}\right)}\right) \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J + J}, \frac{U}{J + J}, 1\right)} \]

       3. lower-*.f64 N/A

          \[\leadsto \left(\left(J + \frac{-1}{8} \cdot \left(J \cdot \color{blue}{{K}^{2}}\right)\right) \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J + J}, \frac{U}{J + J}, 1\right)} \]

       4. lower-pow.f64 37.7%

          \[\leadsto \left(\left(J + -0.125 \cdot \left(J \cdot {K}^{\color{blue}{2}}\right)\right) \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J + J}, \frac{U}{J + J}, 1\right)} \]

   11. Applied rewrites 37.7%

       \[\leadsto \left(\color{blue}{\left(J + -0.125 \cdot \left(J \cdot {K}^{2}\right)\right)} \cdot -2\right) \cdot \sqrt{\mathsf{fma}\left(\frac{U}{J + J}, \frac{U}{J + J}, 1\right)} \]

   if 420 < K

   1. Initial program 73.8%

      \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

   2. Taylor expanded in J around 0

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{\color{blue}{J}} \]

      2. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      3. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      6. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      7. lower-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

      8. lower-*.f64 14.1%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

   4. Applied rewrites 14.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J}} \]

   5. Taylor expanded in U around -inf

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{-1 \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}{J} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{-1 \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}{J} \]

      2. lower-*.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{-1 \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}{J} \]

      3. lower-sqrt.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{-1 \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}{J} \]

      4. lower-/.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{-1 \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}{J} \]

      5. lower-pow.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{-1 \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}{J} \]

      6. lower-cos.f64 N/A

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{-1 \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}{J} \]

      7. lower-*.f64 20.8%

         \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{-1 \cdot \left(U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}\right)}{J} \]

   7. Applied rewrites 20.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{-1 \cdot \left(U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}\right)}{J} \]

   9. Applied rewrites 20.8%

      \[\leadsto \color{blue}{\left(\frac{\left(-U\right) \cdot \frac{\sqrt{0.25}}{\left|\cos \left(K \cdot -0.5\right)\right|}}{J} \cdot \cos \left(K \cdot -0.5\right)\right) \cdot \left(-2 \cdot J\right)} \]

   10. Taylor expanded in J around inf

       \[\leadsto \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)} \cdot \left(-2 \cdot J\right) \]
   11. Step-by-step derivation

       1. lower-cos.f64 N/A

          \[\leadsto \cos \left(\frac{-1}{2} \cdot K\right) \cdot \left(-2 \cdot J\right) \]

       2. lower-*.f64 51.9%

          \[\leadsto \cos \left(-0.5 \cdot K\right) \cdot \left(-2 \cdot J\right) \]

   12. Applied rewrites 51.9%

       \[\leadsto \color{blue}{\cos \left(-0.5 \cdot K\right)} \cdot \left(-2 \cdot J\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 51.9% accurate, 2.6× speedup

Math

\[\cos \left(-0.5 \cdot K\right) \cdot \left(-2 \cdot J\right) \]

FPCore

(FPCore (J K U) :precision binary64 (* (cos (* -0.5 K)) (* -2.0 J)))

C

double code(double J, double K, double U) {
	return cos((-0.5 * K)) * (-2.0 * J);
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(j, k, u)
use fmin_fmax_functions
    real(8), intent (in) :: j
    real(8), intent (in) :: k
    real(8), intent (in) :: u
    code = cos(((-0.5d0) * k)) * ((-2.0d0) * j)
end function

Java

public static double code(double J, double K, double U) {
	return Math.cos((-0.5 * K)) * (-2.0 * J);
}

Python

def code(J, K, U):
	return math.cos((-0.5 * K)) * (-2.0 * J)

Julia

function code(J, K, U)
	return Float64(cos(Float64(-0.5 * K)) * Float64(-2.0 * J))
end

MATLAB

function tmp = code(J, K, U)
	tmp = cos((-0.5 * K)) * (-2.0 * J);
end

Wolfram

code[J_, K_, U_] := N[(N[Cos[N[(-0.5 * K), $MachinePrecision]], $MachinePrecision] * N[(-2.0 * J), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 73.8%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

2. Taylor expanded in J around 0

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J}} \]
3. Step-by-step derivation

   1. lower-/.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{\color{blue}{J}} \]

   2. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

   3. lower-*.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

   4. lower-/.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

   5. lower-pow.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

   6. lower-pow.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

   7. lower-cos.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{\frac{1}{4} \cdot \frac{{U}^{2}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}}{J} \]

   8. lower-*.f64 14.1%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J} \]

4. Applied rewrites 14.1%

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{\frac{\sqrt{0.25 \cdot \frac{{U}^{2}}{{\cos \left(0.5 \cdot K\right)}^{2}}}}{J}} \]

5. Taylor expanded in U around -inf

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{-1 \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}{J} \]
6. Step-by-step derivation

   1. lower-*.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{-1 \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}{J} \]

   2. lower-*.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{-1 \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}{J} \]

   3. lower-sqrt.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{-1 \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}{J} \]

   4. lower-/.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{-1 \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}{J} \]

   5. lower-pow.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{-1 \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}{J} \]

   6. lower-cos.f64 N/A

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{-1 \cdot \left(U \cdot \sqrt{\frac{\frac{1}{4}}{{\cos \left(\frac{1}{2} \cdot K\right)}^{2}}}\right)}{J} \]

   7. lower-*.f64 20.8%

      \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{-1 \cdot \left(U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}\right)}{J} \]

7. Applied rewrites 20.8%

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \frac{-1 \cdot \left(U \cdot \sqrt{\frac{0.25}{{\cos \left(0.5 \cdot K\right)}^{2}}}\right)}{J} \]

9. Applied rewrites 20.8%

   \[\leadsto \color{blue}{\left(\frac{\left(-U\right) \cdot \frac{\sqrt{0.25}}{\left|\cos \left(K \cdot -0.5\right)\right|}}{J} \cdot \cos \left(K \cdot -0.5\right)\right) \cdot \left(-2 \cdot J\right)} \]

10. Taylor expanded in J around inf

    \[\leadsto \color{blue}{\cos \left(\frac{-1}{2} \cdot K\right)} \cdot \left(-2 \cdot J\right) \]
11. Step-by-step derivation

    1. lower-cos.f64 N/A

       \[\leadsto \cos \left(\frac{-1}{2} \cdot K\right) \cdot \left(-2 \cdot J\right) \]

    2. lower-*.f64 51.9%

       \[\leadsto \cos \left(-0.5 \cdot K\right) \cdot \left(-2 \cdot J\right) \]

12. Applied rewrites 51.9%

    \[\leadsto \color{blue}{\cos \left(-0.5 \cdot K\right)} \cdot \left(-2 \cdot J\right) \]
13. Add Preprocessing

Alternative 14: 27.2% accurate, 6.2× speedup

Math

\[\mathsf{fma}\left(-2, J, \left(\left(0.25 \cdot J\right) \cdot K\right) \cdot K\right) \cdot 1 \]

FPCore

(FPCore (J K U)
 :precision binary64
 (* (fma -2.0 J (* (* (* 0.25 J) K) K)) 1.0))

C

double code(double J, double K, double U) {
	return fma(-2.0, J, (((0.25 * J) * K) * K)) * 1.0;
}

Julia

function code(J, K, U)
	return Float64(fma(-2.0, J, Float64(Float64(Float64(0.25 * J) * K) * K)) * 1.0)
end

Wolfram

code[J_, K_, U_] := N[(N[(-2.0 * J + N[(N[(N[(0.25 * J), $MachinePrecision] * K), $MachinePrecision] * K), $MachinePrecision]), $MachinePrecision] * 1.0), $MachinePrecision]

Derivation

1. Initial program 73.8%

   \[\left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \sqrt{1 + {\left(\frac{U}{\left(2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)}\right)}^{2}} \]

2. Taylor expanded in J around inf

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{1} \]
3. Step-by-step derivation

4. Applied rewrites 51.9%

   \[\leadsto \left(\left(-2 \cdot J\right) \cdot \cos \left(\frac{K}{2}\right)\right) \cdot \color{blue}{1} \]

5. Taylor expanded in K around 0

   \[\leadsto \color{blue}{\left(-2 \cdot J + \frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right)} \cdot 1 \]
6. Step-by-step derivation

   1. lower-fma.f64 N/A

      \[\leadsto \mathsf{fma}\left(-2, \color{blue}{J}, \frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right) \cdot 1 \]

   2. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-2, J, \frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right) \cdot 1 \]

   3. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-2, J, \frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right) \cdot 1 \]

   4. lower-pow.f64 27.2%

      \[\leadsto \mathsf{fma}\left(-2, J, 0.25 \cdot \left(J \cdot {K}^{2}\right)\right) \cdot 1 \]

7. Applied rewrites 27.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-2, J, 0.25 \cdot \left(J \cdot {K}^{2}\right)\right)} \cdot 1 \]
8. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-2, J, \frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right) \cdot 1 \]

   2. lift-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-2, J, \frac{1}{4} \cdot \left(J \cdot {K}^{2}\right)\right) \cdot 1 \]

   3. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(-2, J, \left(\frac{1}{4} \cdot J\right) \cdot {K}^{2}\right) \cdot 1 \]

   4. lift-pow.f64 N/A

      \[\leadsto \mathsf{fma}\left(-2, J, \left(\frac{1}{4} \cdot J\right) \cdot {K}^{2}\right) \cdot 1 \]

   5. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(-2, J, \left(\frac{1}{4} \cdot J\right) \cdot \left(K \cdot K\right)\right) \cdot 1 \]

   6. associate-*r* N/A

      \[\leadsto \mathsf{fma}\left(-2, J, \left(\left(\frac{1}{4} \cdot J\right) \cdot K\right) \cdot K\right) \cdot 1 \]

   7. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-2, J, \left(\left(\frac{1}{4} \cdot J\right) \cdot K\right) \cdot K\right) \cdot 1 \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(-2, J, \left(\left(\frac{1}{4} \cdot J\right) \cdot K\right) \cdot K\right) \cdot 1 \]

   9. lower-*.f64 27.2%

      \[\leadsto \mathsf{fma}\left(-2, J, \left(\left(0.25 \cdot J\right) \cdot K\right) \cdot K\right) \cdot 1 \]

9. Applied rewrites 27.2%

   \[\leadsto \mathsf{fma}\left(-2, J, \left(\left(0.25 \cdot J\right) \cdot K\right) \cdot K\right) \cdot 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2025188 
(FPCore (J K U)
  :name "Maksimov and Kolovsky, Equation (3)"
  :precision binary64
  (* (* (* -2.0 J) (cos (/ K 2.0))) (sqrt (+ 1.0 (pow (/ U (* (* 2.0 J) (cos (/ K 2.0)))) 2.0)))))